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Surface Flattening in Garment Design Zhao Hongyan Sep. 13, 2006

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Surface Flattening Application: aircraft industry ship industry shoe industry garment industry

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3D-Computer Aided Garment Design 1. Import several patterns from other 2D garment CAD systems. 2. Obtain 3D garment patterns after a sewing simulation process. 3. Modify the 3D garment patterns by FFD (free- form deformation) tools. 4. Flatten the modified garment patterns.5. 2D comparison

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3D-Computer Aided Garment Design It is important to flatten the modified garment patterns properly, as the modification is always done in the flattened surface in practice.

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Problem Definition Given a 3D freefrom surface and the material properties, find its counterpart pattern in the plane and a mapping relationship between the two so that, when the 2D pattern is folded into the 3D surface, the amount of distortion — wrinkles and stretches — is minimized.

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Measurement of accuracy Area accuracy. A : the actual area of one patch on the surface before development; A ’ : the area of its corresponding patch after development. A can be approximated by summing the area of each triangle in the facet model:

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Measurement of accuracy Shape accuracy. L : the actual length of a curve segment on the original surface before development; L ’ : the corresponding edge length on the developed surface after development. L can be approximated by summing the length of each triangle edge in the facet model:

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Planar parameterization

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Floater 97 ’ Fixing the boundary of the mesh onto a unit circlea unit square

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Planar parameterization For interior mesh points: Forming a sparse linear system

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Surface flattening based on energy model Charlie C.L. Wang, Shana S-F. Smith, Matthew M.F. Yuen CAD 2002;34(11):823-833

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Mass-spring systems ◆ A mass-spring system is established for the deforma- tion of Ф. ◆ Ф is a planar triangular mesh pair (K, P)

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Mass-spring systems

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Elastic deformation energy function Tensile force

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Discrete Lagrange Equation Mass-spring system is governed by Discrete Lagrange Equation:

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Discrete Lagrange Equation f i is the external force. u i is the mass value; r i is the damping coefficient; g i is the total internal force acting on vertex i, due to the spring connections to neighboring vertex j;

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Surface flattening based on energy model Initial triangle flattening Planar mesh deformation

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Surface flattening based on energy model Initial triangle flattening Planar mesh deformation

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◆ Initial triangle flattening ◆ Assume one edge ( Q 1 Q 2 ) has already been flattened.

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◆ Initial triangle flattening §2.1 Unconstrained triangle flattening ◆ The third node ( Q 3 ) is going to be located on the flattened plan.

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◆ Initial triangle flattening(2) # Developable surface # Non-developable surface §2.1 Unconstrained triangle flattening

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◆ Initial triangle flattening(2) §2.2 Constrained triangle flattening When two edges are both available to determine the planar point corresponding to Q3, the obtained two points, shown as P ’ 3 and P ’’ 3, may not be uniform. Original mesh trianglePlanar mesh triangle

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◆ Initial triangle flattening(2.2) §2.2 Constrained triangle flattening In this case, a mean position is used

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Surface flattening based on energy model Initial triangle flattening Planar mesh deformation

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◆ Planar mesh deformation Discrete Lagrange Equation can also be written in the following form: Discrete Lagrange Equation M : spring mass; D : damping matrix; K : stiffness matrix. Ignore the damping item

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◆ Planar mesh deformation For each node P i, the equation can be changed to m i : the mass of P i ; ρ: the area density of the surface; q i ( t ): the position of P i at time t ; f i ( t ): the tensile force on node P i ;tensile force

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◆ Planar mesh deformation Penalty function Goal: t o prevent an overlap

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◆ Planar mesh deformation the deformation process is described by the algorithm in the following

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◆ Examples Example.1 a ruled surface and its 2D patternExample.2 a trimmed surface and its 2D Pattern Table. 1 Calculation statistics of Example. 1 and Example 2

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◆ Additional phase: Initial Energy Release Since energy was generated in the first phase: Constrained triangle flattening, overlapping error would happen.Constrained triangle flattening Original 3D mesh surfaceSurface development without energy release

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◆ Additional phase: Initial Energy Release Therefore, the energy release is added

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◆ Additional phase: Initial Energy Release Original 3D mesh surface Surface development without energy release Surface development with energy release

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◆ Additional phase: Surface Cutting Surface cutting Some complex surfaces difficult to develop

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Surface cutting Firstly, compute the energy on the developed surface. ◆ Additional phase: Surface Cutting

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Surface cutting Second, determine a reference cutting line using an elastic deformation energy distribution map. ◆ Additional phase: Surface Cutting

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Freeform surface flattening based on fitting a woven mesh model Charlie C.L. Wang, Kai Tang, Benjamin M.L. Yeung CAD 2005;37(8):799-814

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Woven mesh model Planar woven fabric Weft / warp springs: tensile-strain resistance Diagonal springs: shear-strain resistance Node V i,j : intersection between springs

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Woven mesh model: assumption 1. The weft threads and the warp threads are not extendable. 2. No slippage occurs at the crossing of a weft and a warp thread. 3. A thread between two adjacent crossing is mapped to a geodesic curve segment on the 3D surface.

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Woven mesh model: assumption The directions of weft and warp springs are orthogonal to each other. Users specify Initial length of springs: r weft, r warp.(r diag ) Center Node: V i C, j C Tendon node: V i,j (i=i c, or j=j c ) Region node: otherwise. Type- I/II/III/IV node

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Strain energy

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Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

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Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

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Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

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Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

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TNM Specify a center point p C and a warp direction vector t warp on M. Compute the weft direction vector t weft. Call Algorithm ComputeDiscreteGeodesicPath(V i C, j C, t warp, M,r warp ). Iteratively until the boundary of M is reached. Determine all the tendon nodes.

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Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

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DNM Four quadrants For a type-I node V i,j (1) Assume V i-1,j-1, V i-1,j and V i,j-1 all have been positioned; (2) Determine two unit vectors and ; (3) Set the diagonal direction as t diag =1/2(t 1 +t 2 ); (4) Staring at V i-1,j-1, search the point on the geodesic path along the t diag direction with distance r diag, by calling Compute Geodesic algorithm iteratively;

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Strain energy release (5) Locally adjust the position of V i,j

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Boundary propagation

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Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

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Energy minimization by diffusion Goal: minimize the strain energythe strain energy Solution: let every node V i satisfy where

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Insertion of darts 1. A specified space curve 2. Delaunay triangulation 3. The fitted woven mesh

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Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

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Surface to plane mapping

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Experiments

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Comparison

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